Simulation Studies for Evaluating ForestSearch Performance

Operating Characteristics and Power Analysis

Author

ForestSearch Package

Published

January 26, 2026

1 Introduction

This vignette demonstrates how to conduct simulation studies to evaluate the performance of ForestSearch for identifying subgroups with differential treatment effects. The simulation framework allows you to:

Generate synthetic clinical trial data with known treatment effect heterogeneity
Evaluate subgroup identification rates (power)
Assess classification accuracy (sensitivity, specificity, PPV, NPV)
Compare different analysis methods (ForestSearch, GRF)
Estimate Type I error under null hypothesis

1.1 Simulation Framework Overview

The simulation workflow consists of four main steps:

flowchart LR
    A[Create DGM] --> B[Simulate Trials]
    B --> C[Run Analyses]
    C --> D[Summarize Results]

Create DGM: Define a data generating mechanism with specified treatment effects
Simulate Trials: Generate multiple simulated datasets
Run Analyses: Apply ForestSearch (and optionally GRF) to each dataset
Summarize Results: Aggregate operating characteristics across simulations

2 Setup

# Core packages
library(forestsearch)
library(weightedsurv)

library(data.table)
library(survival)
library(ggplot2)
library(gt)

# Parallel processing
library(foreach)
library(doFuture)
library(future)

# Source simulation functions (if not yet in package)
# source("sim_aft_gbsg_refactored.R")
# source("oc_analyses_refactored.R")

3 Creating a Data Generating Mechanism

The simulation framework uses the German Breast Cancer Study Group (GBSG) dataset as a template for realistic covariate distributions and censoring patterns.

3.1 Understanding the DGM

The create_gbsg_dgm() function creates a data generating mechanism (DGM) based on an Accelerated Failure Time (AFT) model with Weibull distribution. Key features:

Covariates: Age, estrogen receptor, menopausal status, progesterone receptor, nodes
Treatment effect heterogeneity: Specified via interaction terms
Subgroup definition: H = {low estrogen receptor AND premenopausal}
Censoring: Weibull or uniform censoring model

3.2 Alternative Hypothesis (Heterogeneous Treatment Effect)

Under the alternative hypothesis, we create a DGM where the treatment effect varies across patient subgroups:

# Create DGM with heterogeneous treatment effect
# HR in harm subgroup (H) will be > 1 (treatment harmful)
# HR in complement (H^c) will be < 1 (treatment beneficial)

dgm_alt <- create_gbsg_dgm(
  model = "alt",
  k_treat = 1.0,
  k_inter = 2.0,      # Interaction effect multiplier
  k_z3 = 1.0,
  z1_quantile = 0.25, # ER threshold at 25th percentile
  n_super = 5000,
  cens_type = "weibull",
  seed = 8316951,
  verbose = TRUE
)

# Examine the DGM
print(dgm_alt)

GBSG-Based AFT Data Generating Mechanism
=========================================

Model type: alt 
Super-population size: 5000 

Effect Modifiers:
  k_treat: 1 
  k_inter: 2 
  k_z3: 1 

Hazard Ratios:
  Overall (causal): 0.7094 
  Harm subgroup (H): 3.0741 
  Complement (H^c): 0.6353 
  Ratio HR(H)/HR(Hc): 4.839 

Subgroup definition: z1 == 1 & z3 == 1 (low ER & premenopausal) 
ER threshold: 8 (quantile = 0.25)
Analysis variables: v1, v2, v3, v4, v5, v6, v7

3.3 Calibrating for a Target Hazard Ratio

Often, you want to specify the exact hazard ratio in the harm subgroup. Use calibrate_k_inter() to find the interaction parameter that achieves this:

# Find k_inter for HR = 1.5 in harm subgroup
k_inter_target <- calibrate_k_inter(
  target_hr_harm = 2.0,
  model = "alt",
  k_treat = 1.0,
  cens_type = "weibull",
  verbose = TRUE
)

# Create DGM with calibrated k_inter
dgm_calibrated <- create_gbsg_dgm(
  model = "alt",
  k_treat = 1.0,
  k_inter = k_inter_target,
  verbose = TRUE
)

cat("\nVerification:\n")


Verification:

cat("Achieved HR(H):", round(dgm_calibrated$hr_H_true, 3), "\n")

Achieved HR(H): 2

cat("HR(H^c):", round(dgm_calibrated$hr_Hc_true, 3), "\n")

HR(H^c): 0.635

cat("Overall HR:", round(dgm_calibrated$hr_causal, 3), "\n")

Overall HR: 0.697

3.4 Null Hypothesis (Uniform Treatment Effect)

For Type I error evaluation, create a DGM with uniform treatment effect:

# Create null DGM (no treatment effect heterogeneity)
dgm_null <- create_gbsg_dgm(
  model = "null",
  k_treat = 1.0,
  verbose = TRUE
)

cat("\nNull hypothesis HRs:\n")


Null hypothesis HRs:

cat("Overall HR:", round(dgm_null$hr_causal, 3), "\n")

Overall HR: 0.698

cat("HR(H^c):", round(dgm_null$hr_Hc_true, 3), "\n")

HR(H^c): 0.698

4 Simulating Trial Data

4.1 Single Trial Simulation

Use simulate_from_gbsg_dgm() to generate a single simulated trial:

# Simulate a single trial
sim_data <- simulate_from_gbsg_dgm(
  dgm = dgm_calibrated,
  n = 700,
  rand_ratio = 1,        # 1:1 randomization
  sim_id = 1,
  max_follow = 84,       # 84 months administrative censoring
  muC_adj = log(1.5)            # No censoring adjustment
)

# Examine the data
cat("Simulated trial:\n")

Simulated trial:

cat("  N =", nrow(sim_data), "\n")

  N = 700

cat("  Events =", sum(sim_data$event.sim), 
    "(", round(100 * mean(sim_data$event.sim), 1), "%)\n")

  Events = 374 ( 53.4 %)

cat("  Harm subgroup size =", sum(sim_data$flag.harm),
    "(", round(100 * mean(sim_data$flag.harm), 1), "%)\n")

  Harm subgroup size = 86 ( 12.3 %)

# Quick survival analysis
fit_itt <- coxph(Surv(y.sim, event.sim) ~ treat, data = sim_data)
cat("  Estimated ITT HR =", round(exp(coef(fit_itt)), 3), "\n")

  Estimated ITT HR = 0.731

4.2 Examining Covariate Structure

dfcount <- df_counting(
  df = sim_data,
  by.risk = 6,
  tte.name = "y.sim", 
  event.name = "event.sim", 
  treat.name = "treat"
)
plot_weighted_km(dfcount, conf.int = TRUE, show.logrank = TRUE, ymax = 1.05, xmed.fraction = 0.775, ymed.offset = 0.125)

create_summary_table(data = sim_data, treat_var = "treat", 
                     table_title = "Characteristics by Treatment Arm",
                                      vars_continuous=c("z1","z2","size","z3","z4","z5"),
                                      vars_categorical=c("flag.harm","grade3"),
                                      font_size = 12)

Characteristics by Treatment Arm
Characteristic		Control (n=350)	Treatment (n=350)	P-value¹	SMD²
z1	Mean (SD)	0.3 (0.4)	0.2 (0.4)	0.729	0.03
z2	Mean (SD)	2.4 (1.1)	2.5 (1.1)	0.157	0.11
size	Mean (SD)	29.2 (14.3)	30.1 (17.0)	0.461	0.06
z3	Mean (SD)	0.4 (0.5)	0.4 (0.5)	0.249	0.09
z4	Mean (SD)	2.6 (1.1)	2.5 (1.1)	0.428	0.06
z5	Mean (SD)	2.4 (1.1)	2.4 (1.1)	0.563	0.04
flag.harm				0.908	0.00
	0	306 (87.4%)	308 (88.0%)
	1	44 (12.6%)	42 (12.0%)
grade3				0.530	0.02
	0	273 (78.0%)	265 (75.7%)
	1	77 (22.0%)	85 (24.3%)
¹ P-values: t-test for continuous, chi-square/Fisher's exact for categorical/binary variables
² SMD = Standardized mean difference (Cohen's d for continuous, Cramer's V for categorical)

5 Running Simulation Studies

5.1 Setting Up Parallel Processing

For efficient simulation studies, use parallel processing:

# Configure parallel backend
n_workers <- min(parallel::detectCores() - 1, 120)

plan(multisession, workers = n_workers)
registerDoFuture()

cat("Using", n_workers, "parallel workers\n")

Using 13 parallel workers

5.2 Define Simulation Parameters

# Simulation settings
sim_config_alt <- list(
  n_sims = 1000,          # Number of simulations (use 500-1000 for final)
  n_sample = 700,        # Sample size per trial
  max_follow = 84,       # Maximum follow-up (months)
  seed_base = 8316951,
  muC_adj = log(1.5)
)

sim_config_null <- list(
  n_sims = 1000,          # Number of simulations (use 500-1000 for final)
  n_sample = 700,        # Sample size per trial
  max_follow = 84,       # Maximum follow-up (months)
  seed_base = 8316951,
  muC_adj = log(1.5)
)


# ForestSearch parameters
fs_params <- list(
  outcome.name = "y.sim",
  event.name = "event.sim",
  treat.name = "treat",
  id.name = "id",
  hr.threshold = 1.0,
  hr.consistency = 0.90,
  pconsistency.threshold = 0.90,
  use_grf = TRUE,
  use_lasso = FALSE,
  fs.splits = 400,
  n.min = 60,
  maxk = 2
)

# Confounders for analysis
confounders_base <- c("z1", "z2", "z3", "z4", "z5", "size", "grade3")

5.3 Running the Simulation Loop

# Track timing
t_start <- Sys.time()

sim_config <- sim_config_alt

# Run simulations in parallel
results_alt <- foreach(
  sim = 1:sim_config$n_sims,
  .combine = rbind,
  .errorhandling = "remove",
  .options.future = list(
    packages = c("forestsearch", "survival", "data.table"),
    seed = TRUE
  )
) %dofuture% {
# Run single simulation analysis
run_simulation_analysis(
    sim_id = sim,
    dgm = dgm_calibrated,
    n_sample = sim_config$n_sample,
    max_follow = sim_config$max_follow,
    muC_adj = sim_config$muC_adj,
    confounders_base = confounders_base,
    n_add_noise = 0,
    run_fs = TRUE,
    run_fs_grf = FALSE,
    run_grf = TRUE,      # Set TRUE if grf_subg_harm_survival available
    fs_params = fs_params,
    n_sims_total = sim_config$n_sims,
    seed_base = sim_config$seed_base,
    verbose = TRUE,
    verbose_n = 1
  )
}


=== Two-Stage Consistency Evaluation Enabled ===
Stage 1 screening splits: 30 
Maximum total splits: 400 
Batch size: 20 
================================================

GRF stage for cut selection with dmin, tau = 12 0.6 
tau, maxdepth = 48.53742 2 
   leaf.node control.mean control.size control.se depth
1          2        -5.32       520.00       1.11     1
2          3         0.82       180.00       2.32     1
11         4        -6.70       426.00       1.21     2
21         5         3.23       142.00       2.62     2
4          7        -7.13        78.00       3.18     2
GRF subgroup NOT found
GRF cuts identified: 0 
# of continuous/categorical characteristics 4 3 
Continuous characteristics: z2 z4 z5 size 
Categorical characteristics: z1 z3 grade3 
Default cuts included (1st 20) z2 <= mean(z2) z2 <= median(z2) z2 <= qlow(z2) z2 <= qhigh(z2) z4 <= mean(z4) z4 <= median(z4) z4 <= qlow(z4) z4 <= qhigh(z4) z5 <= mean(z5) z5 <= median(z5) z5 <= qlow(z5) z5 <= qhigh(z5) size <= mean(size) size <= median(size) size <= qlow(size) size <= qhigh(size) 
Categorical: z1 z3 grade3 

===== CONSOLIDATED CUT EVALUATION (IMPROVED) =====
Evaluating 17 cut expressions once and caching...
Cut evaluation summary:
  Total cuts:  17 
  Valid cuts:  16 
  Errors:  0 
Dropping variables (cut only has 1 level): z2 <= 4 
Total cuts after dropping invalid: 16
✓ All 16 factors validated as 0/1
===== END CONSOLIDATED CUT EVALUATION =====

# of candidate subgroup factors= 16 
 [1] "z2 <= 2.5"    "z2 <= 2"      "z4 <= 2.5"    "z4 <= 3"      "z4 <= 2"     
 [6] "z5 <= 2.4"    "z5 <= 2"      "z5 <= 1"      "z5 <= 3"      "size <= 29.1"
[11] "size <= 25"   "size <= 20"   "size <= 35"   "z1"           "z3"          
[16] "grade3"      
Number of possible configurations (<= maxk): maxk = 2 , # combinations = 528 
Events criteria: control >= 12 , treatment >= 12 
Subgroup search completed in 0.02 minutes
Found 34 subgroup candidate(s)
# of candidate subgroups (meeting all criteria) = 34 
Random seed set to: 8316951 
Removed 8 near-duplicate subgroups
Original rows: 34 
After removal: 26 
# of unique initial candidates: 26 
# Restricting to top stop_Kgroups = 10 
# of candidates to evaluate: 10 
Algorithm: Two-stage sequential 
  Stage 1 splits: 30 
  Screen threshold: 0.763 
  Max total splits: 400 
  Batch size: 20 
Parallel processing: callr with 6 workers
SG focus= hr 
Subgroup Consistency Minutes= 0.276 
Algorithm used: Two-stage sequential 
Candidates evaluated: 10 
Candidates passed: 2 
Subgroup found (FS) with sg_focus='hr'
Selected subgroup: {z1} & !{z5 <= 1} 
Minutes forestsearch overall = 0.34 
Consistency algorithm used: twostage 
tau, maxdepth = 48.53742 2 
   leaf.node control.mean control.size control.se depth
1          2        -5.32       520.00       1.11     1
2          3         0.82       180.00       2.32     1
11         4        -6.70       426.00       1.21     2
21         5         3.23       142.00       2.62     2
4          7        -7.13        78.00       3.18     2
GRF subgroup NOT found

# Calculate runtime
t_end <- Sys.time()
runtime_mins <- as.numeric(difftime(t_end, t_start, units = "mins"))

cat("\n=== Simulation Complete ===\n")


=== Simulation Complete ===

cat("Total simulations:", sim_config$n_sims, "\n")

Total simulations: 1000

cat("Successful:", nrow(results_alt) / length(unique(results_alt$analysis)), "\n")

Successful: 1000

cat("Runtime:", round(runtime_mins, 2), "minutes\n")

Runtime: 19.06 minutes

cat("Per simulation:", round(runtime_mins / sim_config$n_sims * 60, 1), "seconds\n")

Per simulation: 1.1 seconds

5.4 Running Null Hypothesis Simulations (Type I Error)

sim_config <- sim_config_null

# Track timing
t_start <- Sys.time()

# Run null hypothesis simulations
results_null <- foreach(
  sim = 1:sim_config$n_sims,
  .combine = rbind,
  .errorhandling = "remove",
  .options.future = list(
    packages = c("forestsearch", "survival", "data.table"),
    seed = TRUE
  )
) %dofuture% {
  
  run_simulation_analysis(
    sim_id = sim,
    dgm = dgm_null,
    n_sample = sim_config$n_sample,
    max_follow = sim_config$max_follow,
    muC_adj = sim_config$muC_adj,
    confounders_base = confounders_base,
    run_fs = TRUE,
    run_fs_grf = FALSE,
    run_grf = TRUE,
    fs_params = fs_params,
    verbose = TRUE,
    verbose_n = 1
  )
}


=== Two-Stage Consistency Evaluation Enabled ===
Stage 1 screening splits: 30 
Maximum total splits: 400 
Batch size: 20 
================================================

GRF stage for cut selection with dmin, tau = 12 0.6 
tau, maxdepth = 47.91247 2 
  leaf.node control.mean control.size control.se depth
2         3        -4.78       695.00       1.00     1
1         4        -5.09       568.00       1.10     2
GRF subgroup NOT found
GRF cuts identified: 0 
# of continuous/categorical characteristics 4 3 
Continuous characteristics: z2 z4 z5 size 
Categorical characteristics: z1 z3 grade3 
Default cuts included (1st 20) z2 <= mean(z2) z2 <= median(z2) z2 <= qlow(z2) z2 <= qhigh(z2) z4 <= mean(z4) z4 <= median(z4) z4 <= qlow(z4) z4 <= qhigh(z4) z5 <= mean(z5) z5 <= median(z5) z5 <= qlow(z5) z5 <= qhigh(z5) size <= mean(size) size <= median(size) size <= qlow(size) size <= qhigh(size) 
Categorical: z1 z3 grade3 

===== CONSOLIDATED CUT EVALUATION (IMPROVED) =====
Evaluating 17 cut expressions once and caching...
Cut evaluation summary:
  Total cuts:  17 
  Valid cuts:  16 
  Errors:  0 
Dropping variables (cut only has 1 level): z2 <= 4 
Total cuts after dropping invalid: 16
✓ All 16 factors validated as 0/1
===== END CONSOLIDATED CUT EVALUATION =====

# of candidate subgroup factors= 16 
 [1] "z2 <= 2.5"    "z2 <= 2"      "z4 <= 2.5"    "z4 <= 3"      "z4 <= 2"     
 [6] "z5 <= 2.4"    "z5 <= 2"      "z5 <= 1"      "z5 <= 3"      "size <= 29.1"
[11] "size <= 25"   "size <= 20"   "size <= 35"   "z1"           "z3"          
[16] "grade3"      
Number of possible configurations (<= maxk): maxk = 2 , # combinations = 528 
Events criteria: control >= 12 , treatment >= 12 
Subgroup search completed in 0.02 minutes
Found 9 subgroup candidate(s)
# of candidate subgroups (meeting all criteria) = 9 
Random seed set to: 8316951 
Removed 2 near-duplicate subgroups
Original rows: 9 
After removal: 7 
# of unique initial candidates: 7 
# Restricting to top stop_Kgroups = 10 
# of candidates to evaluate: 7 
Algorithm: Two-stage sequential 
  Stage 1 splits: 30 
  Screen threshold: 0.763 
  Max total splits: 400 
  Batch size: 20 
Parallel processing: callr with 6 workers
All subgroup evaluations returned NULL
Minutes forestsearch overall = 0.16 
Consistency algorithm used: twostage 
tau, maxdepth = 47.91247 2 
  leaf.node control.mean control.size control.se depth
2         3        -4.78       695.00       1.00     1
1         4        -5.09       568.00       1.10     2
GRF subgroup NOT found

cat("Null simulations completed:", 
    nrow(results_null) / length(unique(results_null$analysis)), "\n")

Null simulations completed: 1000

# Calculate runtime
t_end <- Sys.time()
runtime_mins <- as.numeric(difftime(t_end, t_start, units = "mins"))

cat("\n=== Simulation Complete ===\n")


=== Simulation Complete ===

cat("Total simulations:", sim_config$n_sims, "\n")

Total simulations: 1000

cat("Successful:", nrow(results_null) / length(unique(results_null$analysis)), "\n")

Successful: 1000

cat("Runtime:", round(runtime_mins, 2), "minutes\n")

Runtime: 15.76 minutes

cat("Per simulation:", round(runtime_mins / sim_config$n_sims * 60, 1), "seconds\n")

Per simulation: 0.9 seconds

6 Analyzing Simulation Results

6.1 Summary Statistics

# Summarize alternative hypothesis results
summary_alt <- summarize_simulation_results(
  results = results_alt,
  digits = 2,
  digits_hr = 3
)

# Summarize null hypothesis results  
summary_null <- summarize_simulation_results(
  results = results_null,
  digits = 2,
  digits_hr = 3
)

# Display summaries
cat("=== Alternative Hypothesis (Power) ===\n")

=== Alternative Hypothesis (Power) ===

print(summary_alt)

                    FS     GRF
any.H            0.970   0.720
sensH            0.900   0.840
sensHc           0.980   0.980
ppH              0.850   0.890
ppHc             0.980   0.970
Avg(#H)         84.000  97.000
minH            61.000  60.000
maxH           222.000 224.000
Avg(#Hc)       618.000 631.000
minHc          478.000 476.000
maxHc          700.000 700.000
hat(H*)          2.264   2.083
hat(hat[H])      2.264   2.083
hat(Hc*)         0.635   0.635
hat(hat[Hc])     0.641   0.632
hat(H*)all       2.264   2.083
hat(Hc*)all      0.635   0.635
hat(ITT)all      0.741   0.741
hat(ITTadj)all     NaN     NaN

cat("\n=== Null Hypothesis (Type I Error) ===\n")


=== Null Hypothesis (Type I Error) ===

print(summary_null)

                    FS     GRF
any.H            0.250   0.070
sensH            0.000   0.000
sensHc           1.000   1.000
ppH                NaN     NaN
ppHc             0.860   0.890
Avg(#H)        101.000  77.000
minH            61.000  60.000
maxH           282.000 121.000
Avg(#Hc)       675.000 695.000
minHc          418.000 579.000
maxHc          700.000 700.000
hat(H*)          1.616   1.507
hat(hat[H])      1.616   1.507
hat(Hc*)         0.698   0.698
hat(hat[Hc])     0.669   0.677
hat(H*)all       1.616   1.507
hat(Hc*)all      0.698   0.698
hat(ITT)all      0.700   0.700
hat(ITTadj)all     NaN     NaN

6.2 Formatted Tables with gt

# Create formatted table for alternative hypothesis

tbl_alt <- format_oc_results(
  results = results_alt,
  digits = 2,
  digits_hr = 3
)

tbl_alt

Operating Characteristics Summary
Method	N Sims	Detection	Classification				Hazard Ratios					Size_H_mean	Size_H_min	Size_H_max
Method	N Sims	Detection	Sensitivity	Specificity	PPV	NPV	HR_H_hat	HR_Hc_hat	HR_H_true	HR_Hc_true	HR_ITT	Size_H_mean	Size_H_min	Size_H_max
FS	1000	0.97	0.85	0.98	0.90	0.98	2.264	0.641	2.264	0.635	0.741	84	61	222
GRF	1000	0.71	0.89	0.97	0.84	0.98	2.083	0.632	2.083	0.635	0.741	97	60	224

# Create formatted table for null hypothesis
tbl_null <- format_oc_results(
  results = results_null,
  digits = 2,
  digits_hr =3
)

tbl_null

Operating Characteristics Summary
Method	N Sims	Detection	Classification				Hazard Ratios					Size_H_mean	Size_H_min	Size_H_max
Method	N Sims	Detection	Sensitivity	Specificity	PPV	NPV	HR_H_hat	HR_Hc_hat	HR_H_true	HR_Hc_true	HR_ITT	Size_H_mean	Size_H_min	Size_H_max
FS	1000	0.25	-	0.86	0.00	1.00	1.616	0.669	1.616	0.698	0.700	101	61	282
GRF	1000	0.07	-	0.89	0.00	1.00	1.507	0.677	1.507	0.698	0.700	77	60	121

6.3 Key Operating Characteristics

# Extract key metrics for comparison
methods <- unique(results_alt$analysis)

metrics_comparison <- data.frame(
  Method = methods,
  Power = sapply(methods, function(m) {
    mean(results_alt[results_alt$analysis == m, ]$any.H)
  }),
  TypeI = sapply(methods, function(m) {
    mean(results_null[results_null$analysis == m, ]$any.H)
  }),
  Sensitivity = sapply(methods, function(m) {
    mean(results_alt[results_alt$analysis == m, ]$sensitivity, na.rm = TRUE)
  }),
  PPV = sapply(methods, function(m) {
    mean(results_alt[results_alt$analysis == m, ]$ppv, na.rm = TRUE)
  })
)

metrics_comparison |>
  gt() |>
  tab_header(
    title = "Method Comparison",
    subtitle = sprintf("N = %d, n = %d per trial", 
                       sim_config$n_sims, sim_config$n_sample)
  ) |>
  fmt_percent(columns = c(Power, TypeI, Sensitivity, PPV), decimals = 1) |>
  cols_label(
    Method = "Analysis Method",
    Power = "Power",
    TypeI = "Type I Error",
    Sensitivity = "Sensitivity",
    PPV = "PPV"
  )

Method Comparison
N = 1000, n = 700 per trial
Analysis Method	Power	Type I Error	Sensitivity	PPV
FS	97.4%	25.0%	85.0%	89.9%
GRF	71.5%	6.7%	88.6%	83.9%

7 Visualizing Results

7.1 Subgroup Detection Rates

# Detection rates by method
detection_data <- results_alt[, .(
  detection_rate = mean(any.H),
  se = sqrt(mean(any.H) * (1 - mean(any.H)) / .N)
), by = analysis]

ggplot(detection_data, aes(x = analysis, y = detection_rate, fill = analysis)) +
  geom_col(width = 0.6) +
  geom_errorbar(
    aes(ymin = detection_rate - 1.96 * se, 
        ymax = detection_rate + 1.96 * se),
    width = 0.2
  ) +
  geom_hline(yintercept = 0.80, linetype = "dashed", color = "red", alpha = 0.7) +
  annotate("text", x = 0.6, y = 0.82, label = "80% Power", hjust = 0, size = 3) +
  scale_y_continuous(labels = scales::percent, limits = c(0, 1)) +
  labs(
    x = "Analysis Method",
    y = "Subgroup Detection Rate",
    title = "Power to Detect Harm Subgroup",
    subtitle = sprintf("HR(H) = %.2f, n = %d, %d simulations",
                       dgm_calibrated$hr_H_true, 
                       sim_config$n_sample,
                       sim_config$n_sims)
  ) +
  theme_minimal() +
  theme(legend.position = "none")

7.2 Hazard Ratio Distributions

# Prepare data for plotting
hr_long <- melt(
  results_alt,
  id.vars = c("sim", "analysis", "any.H"),
  measure.vars = c("hr.itt", "hr.H.hat", "hr.Hc.hat"),
  variable.name = "estimand",
  value.name = "hr"
)

hr_long$estimand <- factor(hr_long$estimand, 
  levels = c("hr.itt", "hr.H.hat", "hr.Hc.hat"),
  labels = c("ITT", "Harm Subgroup (H)", "Complement (H^c)")
)

# True values for reference
true_hrs <- data.frame(
  estimand = c("ITT", "Harm Subgroup (H)", "Complement (H^c)"),
  true_hr = c(dgm_calibrated$hr_causal, 
              dgm_calibrated$hr_H_true, 
              dgm_calibrated$hr_Hc_true)
)

ggplot(hr_long[!is.na(hr_long$hr), ], 
       aes(x = estimand, y = hr, fill = analysis)) +
  geom_violin(position = position_dodge(0.8), alpha = 0.7) +
  geom_boxplot(position = position_dodge(0.8), width = 0.15, 
               outlier.size = 0.5, alpha = 0.9) +
  geom_hline(data = true_hrs, aes(yintercept = true_hr), 
             linetype = "dashed", color = "red") +
  geom_hline(yintercept = 1, linetype = "solid", color = "gray50", alpha = 0.5) +
  scale_y_log10() +
  labs(
    x = "Estimand",
    y = "Hazard Ratio (log scale)",
    fill = "Method",
    title = "Distribution of Hazard Ratio Estimates",
    subtitle = "Dashed red lines indicate true values"
  ) +
  theme_minimal() +
  theme(axis.text.x = element_text(angle = 15, hjust = 1))

7.3 Classification Performance

# Classification metrics
class_metrics <- results_alt[any.H == 1, .(
  Sensitivity = mean(sensitivity, na.rm = TRUE),
  Specificity = mean(specificity, na.rm = TRUE),
  PPV = mean(ppv, na.rm = TRUE),
  NPV = mean(npv, na.rm = TRUE)
), by = analysis]

class_long <- melt(class_metrics, id.vars = "analysis", 
                   variable.name = "metric", value.name = "value")

ggplot(class_long, aes(x = metric, y = value, fill = analysis)) +
  geom_col(position = position_dodge(0.8), width = 0.7) +
  geom_hline(yintercept = 0.8, linetype = "dashed", alpha = 0.5) +
  scale_y_continuous(labels = scales::percent, limits = c(0, 1)) +
  labs(
    x = "Classification Metric",
    y = "Value",
    fill = "Method",
    title = "Classification Performance (When Subgroup Found)",
    subtitle = "Sensitivity = P(in Ĥ | in H), PPV = P(in H | in Ĥ)"
  ) +
  theme_minimal()

7.4 Subgroup Size Distribution

# Size distribution when subgroup found
size_data <- results_alt[any.H == 1]

ggplot(size_data, aes(x = size.H, fill = analysis)) +
  geom_histogram(position = "identity", alpha = 0.6, bins = 30) +
  geom_vline(xintercept = sim_config$n_sample * mean(dgm_calibrated$df_super_rand$flag.harm),
             linetype = "dashed", color = "red") +
  facet_wrap(~ analysis, ncol = 1) +
  labs(
    x = "Estimated Subgroup Size",
    y = "Count",
    title = "Distribution of Estimated Subgroup Sizes",
    subtitle = "Dashed line = expected true subgroup size"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

8 Theoretical Subgroup Detection Rate Approximation

The function compute_detection_probability() provides an analytical approximation based on asymptotic normal theory:

#| label: theoretical-power
#| fig-width: 8
#| fig-height: 6

# # Quick check using older code
# hr1 <- 1.25
# hr2 <- 1.0
# library(cubature)
# # Mild censoring
# n.sg <- 60
# pC <- 0.20
# hrH.plims<-seq(0.5,3.0,length=60)
# hrH.plims <- sort(c(hrH.plims,0.5,0.7,0.75,0.80))
# 
# pAnyH.approx <-rep(NA,length(hrH.plims))
# for(hh in 1:length(hrH.plims)){
#   hrH.plim <- hrH.plims[hh]
#   pAnyH.approx[hh] <- compute_detection_probability(
#  theta = hrH.plim,
#   n_sg = n.sg,
#   prop_cens = pC,
#   hr_threshold = hr1,
#   hr_consistency = hr2,
#   method = "cubature"
# )
# }
# plot(hrH.plims,pAnyH.approx,xlab="Hazard ratio for subgroup H",ylab="Probability of any H found",type="l",lty=1,lwd=2.0,ylim=c(0,1))
# abline(h=0.10,lwd=0.5,col="red")
# Looks good!

# =============================================================================
# Theoretical Detection Probability Analysis
# =============================================================================

# Calculate expected subgroup characteristics
n_sg_expected <- sim_config_alt$n_sample * mean(dgm_calibrated$df_super_rand$flag.harm)
prop_cens <- mean(results_alt$p.cens)  # Censoring proportion

cat("=== Subgroup Characteristics ===\n")

=== Subgroup Characteristics ===

cat("Expected subgroup size (n_sg):", round(n_sg_expected), "\n")

Expected subgroup size (n_sg): 89

cat("Censoring proportion:", round(prop_cens, 3), "\n")

Censoring proportion: 0.453

cat("True HR in H:", round(dgm_calibrated$hr_H_true, 3), "\n")

True HR in H: 2

cat("HR threshold:", fs_params$hr.threshold, "\n")

HR threshold: 1

# -----------------------------------------------------------------------------
# Single-Point Detection Probability
# -----------------------------------------------------------------------------

# True H is dgm_calibrated$hr_H_true
# However we want at plim of observed estimate
#plim_hr_hatH <- c(summary_alt[c("hat(hat[H])"),1])

dgm_calibrated$hr_H_true

   treat 
2.000339

# Compute detection probability at the true HR
prob_detect <- compute_detection_probability(
 theta = dgm_calibrated$hr_H_true,
  n_sg = round(n_sg_expected),
  prop_cens = prop_cens,
  hr_threshold = fs_params$hr.threshold,
  hr_consistency = fs_params$hr.consistency,
  method = "cubature"
)

# Compare theoretical to empirical (alternative)
cat("\n=== Detection Probability Comparison ===\n")


=== Detection Probability Comparison ===

cat("Theoretical FS (asymptotic):", round(prob_detect, 3), "\n")

Theoretical FS (asymptotic): 0.933

cat("Empirical FS:", round(mean(results_alt[analysis == "FS"]$any.H), 3), "\n")

Empirical FS: 0.974

cat("Empirical FSlg:", round(mean(results_alt[analysis == "FSlg"]$any.H), 3), "\n")

Empirical FSlg: NaN

if ("GRF" %in% results_alt$analysis) {
  cat("Empirical GRF:", round(mean(results_alt[analysis == "GRF"]$any.H), 3), "\n")
}

Empirical GRF: 0.715

# Null 

#plim_hr_itt <- c(summary_alt[c("hat(ITT)all"),1])

# Calculate at min SG size
# Compute detection probability at the true HR
prob_detect_null <- compute_detection_probability(
 theta = dgm_null$hr_causal,
  n_sg = fs_params$n.min,
  prop_cens = prop_cens,
  hr_threshold = fs_params$hr.threshold,
  hr_consistency = fs_params$hr.consistency,
  method = "cubature"
)


# Compare theoretical to empirical (alternative)
cat("\n=== Detection Probability Comparison ===\n")


=== Detection Probability Comparison ===

cat("Under the null calculate at min SG size:", fs_params$n.min,"\n")

Under the null calculate at min SG size: 60

cat("Theoretical FS at min(SG) (asymptotic):", round(prob_detect_null, 6), "\n")

Theoretical FS at min(SG) (asymptotic): 0.068463

cat("Empirical FS:", round(mean(results_null[analysis == "FS"]$any.H), 6), "\n")

Empirical FS: 0.25

cat("Empirical FSlg:", round(mean(results_null[analysis == "FSlg"]$any.H), 6), "\n")

Empirical FSlg: NaN

if ("GRF" %in% results_null$analysis) {
  cat("Empirical GRF:", round(mean(results_null[analysis == "GRF"]$any.H), 6), "\n")
}

Empirical GRF: 0.067

# -----------------------------------------------------------------------------
# Generate Full Detection Curve
# -----------------------------------------------------------------------------

# Generate detection probability curve across HR values
detection_curve <- generate_detection_curve(
  theta_range = c(0.5, 3.0),
  n_points = 50,
  n_sg = round(n_sg_expected),
  prop_cens = prop_cens,
  hr_threshold = fs_params$hr.threshold,
  hr_consistency = fs_params$hr.consistency,
  include_reference = TRUE,
  verbose = FALSE
)

# -----------------------------------------------------------------------------
# Visualization
# -----------------------------------------------------------------------------

# Plot detection curve with empirical overlay
plot_detection_curve(
  detection_curve,
  add_reference_lines = TRUE,
  add_threshold_line = TRUE,
  title = sprintf(
    "Detection Probability Curve (n=%d, cens=%.0f%%, threshold=%.2f)",
    round(n_sg_expected), 100 * prop_cens, fs_params$hr.threshold
  )
)

# Add empirical results as points
empirical_rates <- c(
  FS = mean(results_alt[analysis == "FS"]$any.H),
  FSlg = mean(results_alt[analysis == "FSlg"]$any.H)
)
if ("GRF" %in% results_alt$analysis) {
  empirical_rates["GRF"] <- mean(results_alt[analysis == "GRF"]$any.H)
}

# Mark the true HR and empirical detection rates
points(
  x = rep(dgm_calibrated$hr_H_true, length(empirical_rates)),
  y = empirical_rates,
  pch = c(16, 17, 18)[1:length(empirical_rates)],
  col = c("blue", "darkgreen", "purple")[1:length(empirical_rates)],
  cex = 1.5
)

# Add vertical line at true HR
abline(v = dgm_calibrated$hr_H_true, lty = 2, col = "blue", lwd = 1)

# Legend for empirical points
legend(
  "topleft",
  legend = c(
    sprintf("H true = %.2f", dgm_calibrated$hr_H_true),
    paste(names(empirical_rates), "=", round(empirical_rates, 3))
  ),
  pch = c(NA, 16, 17, 18)[1:(length(empirical_rates) + 1)],
  lty = c(2, rep(NA, length(empirical_rates))),
  col = c("blue", "blue", "darkgreen", "purple")[1:(length(empirical_rates) + 1)],
  cex = 0.8,
  bty = "n"
)

# -----------------------------------------------------------------------------
# Compare Across Sample Sizes (Optional)
# -----------------------------------------------------------------------------

# Show how detection probability changes with sample size
cat("\n=== Detection Probability by Sample Size ===\n")


=== Detection Probability by Sample Size ===

cat(sprintf("At true HR = %.2f:\n", dgm_calibrated$hr_H_true))

At true HR = 2.00:

for (n_mult in c(0.5, 1.0, 1.5, 2.0)) {
  n_test <- round(n_sg_expected * n_mult)
  prob_test <- compute_detection_probability(
    theta = dgm_calibrated$hr_H_true,
    n_sg = n_test,
    prop_cens = prop_cens,
    hr_threshold = fs_params$hr.threshold,
    hr_consistency = fs_params$hr.consistency
  )
  cat(sprintf("  n_sg = %3d (%.1fx): P(detect) = %.3f\n", 
              n_test, n_mult, prob_test))
}

  n_sg =  44 (0.5x): P(detect) = 0.811
  n_sg =  89 (1.0x): P(detect) = 0.933
  n_sg = 133 (1.5x): P(detect) = 0.974
  n_sg = 178 (2.0x): P(detect) = 0.990

9 Advanced Topics

9.1 Adding Noise Variables

Test ForestSearch robustness by including irrelevant noise variables:

# Run simulations with noise variables
results_noise <- foreach(
  sim = 1:sim_config$n_sims,
  .combine = rbind,
  .errorhandling = "remove",
  .options.future = list(
    packages = c("forestsearch", "survival", "data.table"),
    seed = TRUE
  )
) %dofuture% {
  
  run_simulation_analysis(
    sim_id = sim,
    dgm = dgm_calibrated,
    n_sample = sim_config$n_sample,
    confounders_base = confounders_base,
    n_add_noise = 10,  # Add 10 noise variables
    run_fs = TRUE,
    fs_params = fs_params,
    verbose = FALSE
  )
}

# Compare detection rates
cat("Without noise:", round(mean(results_alt$any.H), 3), "\n")
cat("With 10 noise vars:", round(mean(results_noise$any.H), 3), "\n")

9.2 Sensitivity Analysis: Varying Parameters

# Test different HR thresholds
thresholds <- c(1.10, 1.25, 1.50)
results_by_thresh <- list()

for (thresh in thresholds) {
  results_by_thresh[[as.character(thresh)]] <- foreach(
    sim = 1:100,
    .combine = rbind,
    .options.future = list(
      packages = c("forestsearch", "survival", "data.table"),
      seed = TRUE
    )
  ) %dofuture% {
    
    run_simulation_analysis(
      sim_id = sim,
      dgm = dgm_calibrated,
      n_sample = sim_config$n_sample,
      confounders_base = confounders_base,
      run_fs = TRUE,
      fs_params = modifyList(fs_params, list(hr.threshold = thresh)),
      verbose = FALSE
    )
  }
  results_by_thresh[[as.character(thresh)]]$threshold <- thresh
}

# Combine and summarize
combined <- rbindlist(results_by_thresh)
combined[, .(power = mean(any.H), ppv = mean(ppv, na.rm = TRUE)), 
         by = .(threshold, analysis)]

10 Saving Results

# Save simulation results for later use
save_simulation_results(
  results = results_alt,
  dgm = dgm_calibrated,
  summary_table = summary_alt,
  runtime_hours = runtime_mins / 60,
  output_file = "forestsearch_simulation_alt.Rdata",
  power_approx = power_approx
)

save_simulation_results(
  results = results_null,
  dgm = dgm_null,
  summary_table = summary_null,
  runtime_hours = runtime_mins / 60,
  output_file = "forestsearch_simulation_null.Rdata"
)

11 Complete Example Script

Here’s a minimal self-contained script for running a simulation study:

# ===========================================================================
# Complete ForestSearch Simulation Study - Minimal Example
# ===========================================================================

library(forestsearch)
library(data.table)
library(survival)
library(foreach)
library(doFuture)

# --- Configuration ---
N_SIMS <- 500
N_SAMPLE <- 500
TARGET_HR_HARM <- 1.5

# --- Setup parallel processing ---
plan(multisession, workers = 4)
registerDoFuture()

# --- Create DGM ---
k_inter <- calibrate_k_inter(target_hr_harm = TARGET_HR_HARM, verbose = TRUE)
dgm <- create_gbsg_dgm(model = "alt", k_inter = k_inter, verbose = TRUE)

# --- Run simulations ---
confounders <- c("z1", "z2", "z3", "z4", "z5", "size", "grade3")

results <- foreach(
  sim = 1:N_SIMS, 
  .combine = rbind,
  .options.future = list(
    packages = c("forestsearch", "survival", "data.table"),
    seed = TRUE
  )
) %dofuture% {
  run_simulation_analysis(
    sim_id = sim,
    dgm = dgm,
    n_sample = N_SAMPLE,
    max_follow = 60,
    confounders_base = confounders,
    run_fs = TRUE,
    run_fs_grf = TRUE,
    run_grf = FALSE,
    fs_params = list(hr.threshold = 1.25, fs.splits = 300, maxk = 2)
  )
}

# --- Summarize ---
summary_table <- summarize_simulation_results(results)
print(summary_table)

# --- Display formatted table ---
format_oc_results(summary_table, n_sims = N_SIMS, model_type = "alt")

12 Summary

This vignette demonstrated the complete workflow for evaluating ForestSearch performance through simulation:

Step	Function	Purpose
1. Create DGM	`create_gbsg_dgm()`	Define data generating mechanism
2. Calibrate	`calibrate_k_inter()`	Achieve target subgroup HR
3. Simulate	`simulate_from_gbsg_dgm()`	Generate trial data
4. Analyze	`run_simulation_analysis()`	Run ForestSearch/GRF
5. Summarize	`summarize_simulation_results()`	Aggregate metrics
6. Display	`format_oc_results()`	Create gt tables

Key metrics to report:

Power (H1) / Type I Error (H0): Subgroup detection rate
Sensitivity: P(identified | true harm subgroup)
Specificity: P(not identified | true complement)
PPV: P(true harm | identified)
NPV: P(true complement | not identified)

13 Session Info

sessionInfo()

R version 4.5.1 (2025-06-13)
Platform: aarch64-apple-darwin20
Running under: macOS Tahoe 26.2

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Los_Angeles
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] doFuture_1.1.2          future_1.67.0           foreach_1.5.2          
[4] gt_1.1.0                ggplot2_4.0.0           survival_3.8-3         
[7] data.table_1.17.8       weightedsurv_0.1.0      forestsearch_0.0.0.9000

loaded via a namespace (and not attached):
 [1] sass_0.4.10          generics_0.1.4       xml2_1.4.0          
 [4] shape_1.4.6.1        stringi_1.8.7        lattice_0.22-7      
 [7] cubature_2.1.4       listenv_0.9.1        digest_0.6.37       
[10] magrittr_2.0.4       grf_2.5.0            evaluate_1.0.5      
[13] grid_4.5.1           RColorBrewer_1.1-3   iterators_1.0.14    
[16] policytree_1.2.3     fastmap_1.2.0        jsonlite_2.0.0      
[19] Matrix_1.7-3         glmnet_4.1-10        scales_1.4.0        
[22] codetools_0.2-20     cli_3.6.5            rlang_1.1.6         
[25] parallelly_1.45.1    future.apply_1.20.0  splines_4.5.1       
[28] withr_3.0.2          yaml_2.3.10          tools_4.5.1         
[31] parallel_4.5.1       dplyr_1.1.4          globals_0.18.0      
[34] vctrs_0.6.5          R6_2.6.1             lifecycle_1.0.4     
[37] stringr_1.5.2        randomForest_4.7-1.2 fs_1.6.6            
[40] htmlwidgets_1.6.4    pkgconfig_2.0.3      progressr_0.17.0    
[43] pillar_1.11.1        gtable_0.3.6         glue_1.8.0          
[46] Rcpp_1.1.0           xfun_0.53            tibble_3.3.0        
[49] tidyselect_1.2.1     rstudioapi_0.17.1    knitr_1.50          
[52] farver_2.1.2         htmltools_0.5.8.1    labeling_0.4.3      
[55] rmarkdown_2.30       compiler_4.5.1       S7_0.2.0

14 References

León LF, Marceau-West CT, He W, et al. (2024). “Identifying Patient Subgroups with Differential Treatment Effects: A Forest Search Approach.” Statistics in Medicine.
Athey S, Imbens GW. (2016). “Recursive partitioning for heterogeneous causal effects.” PNAS, 113(27):7353-7360.
Wager S, Athey S. (2018). “Estimation and inference of heterogeneous treatment effects using random forests.” JASA, 113(523):1228-1242.